
Boolean
Algebra 



Logic gates and circuits 
The
"AND" gate, the "OR" gate and the "NOT"
gate 
Binary number system 
Binary
to decimal conversion 
Decimal
to binary conversion 






Logic
Gates and Circuits 
There
are exactly three basic electronic circuits called logic gates
each of which correspond to one of the three Boolean (binary) operators, “and,” “or,” and
“not” having the same properties. 
AND
gate 
a 
b 
a
·
b 
0 
0 
0 
1 
0 
0 
0 
1 
0 
1 
1 
1 

OR
gate 
a 
b 
a
+ b 
0 
0 
0 
1 
0 
1 
0 
1 
1 
1 
1 
1 

NOT gate (or
invertor) 






Logic
circuits used in digital computers are built up from logic
gates. We want to know the output y
of a logic
circuit for all possible combinations of input bits.
The value of the output is shown at the resultant column of the
corresponding truth table. 

Binary
Number System 
The
binary number system uses digits, 0 and 1 to represent numbers.
A binary number can be therefore represented by any sequence of
bits
(binary digits). 
The tables
show binary representations of integers from 0 to 19 with corresponding place values of bits. 
Decimal
number 
Binary
number 
2^{4} 
2^{3}

2^{2} 
2^{1}

2^{0} 
0 
0 
0 
0 
0 
0 
1 
0 
0 
0 
0 
1 
2 
0 
0 
0 
1 
0 
3 
0 
0 
0 
1 
1 
4 
0 
0 
1 
0 
0 
5 
0 
0 
1 
0 
1 
6 
0 
0 
1 
1 
0 
7 
0 
0 
1 
1 
1 
8 
0 
1 
0 
0 
0 
9 
0 
1 
0 
0 
1 


Decimal
number 
Binary
number 
2^{4} 
2^{3}

2^{2} 
2^{1}

2^{0} 
10 
0 
1 
0 
1 
0 
11 
0 
1 
0 
1 
1 
12 
0 
1 
1 
0 
0 
13 
0 
1 
1 
0 
1 
14 
0 
1 
1 
1 
0 
15 
0 
1 
1 
1 
1 
16 
1 
0 
0 
0 
0 
17 
1 
0 
0 
0 
1 
18 
1 
0 
0 
1 
0 
19 
1 
0 
0 
1 
1 



Binary
to decimal conversion: Any binary number can be converted to its
decimal equivalent by writing it in a placevalue notation, i.e. as the sum of products of each
digit with place value of that digit. 

Example: 
=
1 · 2^{6} + 0
· 2^{5}
+ 1 · 2^{4} + 1 · 2^{3}
+ 1 · 2^{2} + 0 ·
2^{1}
+ 1 · 2^{0} = 

= 64 + 0
+ 16 + 8
+ 4 +
0 + 1
= 93 

Decimal
to binary conversion: 
To convert a decimal number to its
binary equivalent divide given decimal and each successive
quotient by 2 noting remainders from right to left, that is from
the lowest place value to the higher. 
The remainders can only be 0 and 1 since divisions are by 2. The
division ends by the quotient
zero. 


113 
÷ 
2 
= 
1
1 1 1 0 0 0 1 
56 



<==== 
28 




14 




7 




3 




1 




0 














Beginning
Algebra Contents 



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